Physics 262 GMUiitGeorge Mason n versity Prof. Paul So Chapter 35: Interference If d? Interference an Coherent Sources ? Two-Source Interference of Light ? Intensity of Interference Patterns ? Interference in Thin Films ? The Michelson Interferometer Wave Nature of Light Pi Ch (G iOi)? L? Previous Chapters (Geometr c pt cs << ? Rays Model is an approximation of EM waves with rays pointing in the direction of propagation ? Next Couple of Chapters (Wave/Physical Optics) ? ~ L ? Like water waves, light spreads and interferes with each other. ? Observed phenomena cannot be accounted for by rays:yy Diffraction Interference spreading constructive/ destructive interference patterns Interference and Superposition ? It f ft ittiihihtInter erence refers to an situation n whic wo or more waves overlap in space. ? The resultant displacement at any point is governed by the principle of superposition. ?the resultant disturbance at any point and at any instant is found by adding the instantaneous disturbance that would be produced at the point by the individual waves as if each waves was present alone.? Interference and Superposition CiIf(k k) wave 1 Constructive nter erence + pea s < > + peaks) + = wave 2 Destructive Interference (+ peaks < > - peaks: ?/2 apart) + = wave 1 wave 2 Conditions for Observable Sustained Interference Th h b ht1. e sources ave to e coherent ? Th idiid l t iti ttThe individua waves mus ma n a n a constant phase relationship (oscillate in unison) with each other. - e.g. two speakers driven by the same amplifier - two regular light bulbs don?t interfere since they are not h (E i i f li h b lb i f hlco erent. (Em ss on rom a g t u s rom a thermal process of random motions of charged particles in the filament.) Conditions for Observable Sustained Interference Th h ld h h l i i2. e waves s ould ave t e same po ar zat on. 3. Two or more interfering waves must have the same wavelength (monochromatic) (monochromatic) ? You can have white light interference pattern (if the source is coherent) but the effect will appear for )pp different colors corresponding to the different wavelengths in white light. Two-Source Interference of Light D two sheets (a) Point a is symmetric with respect to the two coherent sources. Waves will arrive in phase constructively: r 2 - r 1 = 0. 11 : distance to Sr Constructive Inter. Destructive Inter. 22 21 : distance to Sr rr??path difference 21 ( 0,1,2, ) rrm m ??? ???? 21 1 2 ( 0,1,2, ) rr m m ? ?? ?? ? ?? ?? ???? Young?s Double Slit Experiment Spreading of light behind slits Recall Huygen?s Principle D 2 slits Young?s Double Slit Experiment If screen is far away so that R >> d we can assume rays from S and S to be away so that , 1 2 approximately parallel. Then, from the simplified geometry (right panel), we have an explicit expression for the path difference: the 21 sinrrd ?? ? ?? is the angular location of observation point P on the screen.) Constructive/Destructive Two-Slit Interference Applying the conditions for constructive/ destructive interference, we have the following conditions: Constructive Interference: Two Slit Interference sin ( 0, 1, 2, )dmm? ????? Destructive Interference: Two Slit Interference 1 sin ( 0, 1, 2, ) 2 dm m?? ?? ?? ??? ?? ?? ? Interference: Tw Slit ? The bright/dark bands in the pattern are called fringes ? m is the order of the fringes Locating Fringes d P ? m In normal situations, 1 tth f Rm d ? ? ? ? ? y m R tenths o mm tenths of m?? ? ? ? (400-700 nm) So, typically, we have the condition that d << L so that ? is small. Thus we can approximate sin tan????? , The linear distance to a particular ordered fringe (y m ) is given by: tan mm yR?? Wi h h ll l i i ht t e small ang es approx mat on, we ave: tan sin mmm m yR R R d ? ????? Example 35.1 Determine the wavelength of the light from location of y 3 . ? ? 3 37 3 39.4910 0 2 10 6 33 10 633 y m Rd ? ? ? ? ?? 3 .210 .33 10 3 3 1.00 yR d mmnm dR ? ? ????? ??? Intensity of Interference Patterns Ph R i f E Fi ldasor epresentat on o an e : E ? phasor E(t) ?t E o t E(t) () cos( ) o Et E t?? Efi ld t ( h )ttiith l ith lE ? - E field as a vec or p asor) rotating n the x-y p ane w an angu ar frequency ?. - The time variation of this E field, E(t) is given as the horizontal projection (light blue) of the phasor (dark blue).E ? )p Phasor in Action Efi ld t ( h )ttiith l ith l Copyright George Watson, Univ. of Delaware, 1997 - E field as a vec or p asor) rotating n the x-y p ane w an angu ar frequency ?. - The time variation of this E field, E(t) is given as the vertical projection (light red) of the phasor (dark red).)p Intensity of Interference Patterns Now, we consider the E fields coming from the double slits: S 21 rr? E field from S 2 has a phase lag ? due to the extra path difference, r 2 - r 1 . 2 S 1 2 () cos( )Et E t? ?? ? 1 () cos( )Et E t?? Phase Difference relates to Path Difference Here, we have the lighter cyan wave slightly ahead of the blue wave. r 2 ?r 1 (path difference measured in length) ? (phase difference measured in radians) ? (a complete cycle measured in wavelength) ?? (a complete cycle measured in phase) This gives the relation, 21 rr? ? ? ? ? ? ? ? 21 21 2 rr krr ? ? ? ? ?? ? g, 2?? where k = 2??? is called the wave number. Phase Difference depends on Path Difference Recall from our geometry, we have the following picture for following for the path difference: 21 sinrrd ??? ? is the angular position of the observation point P Substituting this into our previous equation, we have:pq ?? 2 1 22 sin d rr ? ? ? ? ?? ??? Intensity of Interference Patterns At a point P on the screen far away from the two slits, the total E field at PE is given - at P, E P , by the vector-sum of the two phasors . 12 andEE ?? 2 E ? To find the magnitude of the resultant phasor , E P , we hl f i P E ? P E ? ? ?? use the law o cos nes . 222 2 2cos() P EEE E ????? ? 1 E ? E ? E 2 1 Intensity of Interference Patterns Using the trig identity, cos( ) cos? ??? ?? 222 2 222 2cos( ) 22cos P P EEE E EEE ? ? ? ??? ? ?? we have, 22 2(1cos) P EE ??? Using another trig identity, ? ? 2 1cos 2cos 2??? ? 222 4cos 2 P EE ? ?? ? ?? ?? ggy, we have, . 2cos 2 P EE ? ?? ? ?? ?? This gives, . Administrator Microsoft PowerPoint - ch35_young_freedman.ppt [Compatibility Mode]